Calculate $\lim_{x \to 0} \frac{\cos(x)}{\sin(x)}$ I have a question regarding the following limit calculation:
$\lim_{x \to 0} \frac{\cos(x)}{\sin(x)}$
The only way I can solve this is by looking at the one-sided limits:
$\\$ From above:
$\lim_{x \to 0^{+}} \frac{\cos(x)}{\sin(x)}$.
The numerator approaches $1$ with a positive sign. The denominator approaches $0$ with a positive sign.  $\implies$ the limit is $\infty$
$\\$ From below:
$\lim_{x \to 0^{-}} \frac{\cos(x)}{\sin(x)}$.
The numerator approaches $1$ with a positive sign. The denominator approaches $0$ with a negative sign.  $\implies$ the limit is $-\infty$
The one-sided limits do not agree and so the limit does not exist.
My concern is this: would you give full marks for an answer like this? It feels very informal but I do not know how to argue the same thing formally.
 A: I would give you full score without any hesitation. Some professors would ask for an $\epsilon$-$\delta$ proof but in my opinion this would be counter productive. Here are some reasons why I like your reasoning:

*

*You paired the problem to its basic, each sub-goal you have is easy, then you just have to assemble the pieces together and you did tell how you do it.


*Computing the limit of all the terms that are in your expression and then saying that taking a fraction preserves the limit (when the limiting numbers are fractions-friendly) is way more natural than guessing the limit (with your process!) and parachute an $\epsilon$-$\delta$ proof making it "formal".
However if you have any doubt about your proof then you should be more precise and rigorous, which does not necessarily mean performing an $\epsilon$-$\delta$ proof. For instance if you do not trust the tools you used: prove them. Proving the two limit theorems you used is way better than parachuting an $\epsilon$-$\delta$ proof. For mathematics I think a key habit to learn something is to prove everything until you are convinced and without any remaining doubt.
Disclaimer: this is my opinion and I am not a teacher/professor, only a student between first and second year of Master of mathematics.
